As far as the relevance of the off-diagonal elements of \(\mathbf {\alpha }_1\) , we can point to the generalized significance of the parameters that feed into the absolute returns from the other measures (the \(\alpha _{1,j,1}\) 's, \(j=2,3\) , the only exception is for KS11 for realized volatility); the same is true for the \(\alpha _{2,j,1}\) 's, \(j=1,3\) , with an exception for the coefficient from lagged absolute returns for STOXX50E; by contrast, the coefficients \(\alpha _{3,j,1}\) 's, \(j=1,2\)  are generally not significant with some rare exceptions (both for FCHI and the coefficient from lagged realized volatility for N225). These results point to the autonomous, so–to–speak, dynamics of the implied volatility, but also (and foremost) to a full interdependence in what concerns the other two measures. As a matter of fact, if one were to extend the logic of the HEAVY model by [1]} to this framework, the question would be whether absolute returns can be considered irrelevant in their own dynamics (while keeping the lagged dependence on the other measures) and also in the dynamics of the other two measures. We address this question formally by jointly testing the significance of the four parameters \(\alpha _{1,1,1}\) , \(\gamma _{1,1,1}\) , \(\alpha _{2,1,1}\) , \(\alpha _{3,1,1}\) , obtaining the results in the first panel of Table REF . For the \(\mathsf {vMEM}\)  these implied `HEAVY' restrictions are strongly rejected, with the DJI providing significant evidence at 5% but not at 1%.
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